= Derivative of a power of the Lp norm
{title2=$\frac{d}{dt}\|u+tv\|_p^p\big|_{t=0}=p\operatorname{Re}\int|u|^{p-2}\overline u\,v\,d\mu$}
For $1<p<\infty$ and $u,v$ in an <Lp space>, the derivative with respect to the real scalar $t$ has the displayed form; the integrand is zero where $u=0$. The pointwise <chain rule> gives the derivative, while the <Holder inequality> makes $(|u|+|v|)^{p-1}|v|$ integrable and permits the <dominated convergence theorem>. This variational derivative characterizes nearest points to <closed> <convex sets> and constructs the <Lp duality> density of a <bounded linear functional>.
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